FRM Exam Part I · Bond Yields and Return Calculations

Yield Measures and Compounding for FRM Part I

Updated 11 October 2026 · Fact-checked

A yield measure expresses a bond's return as an annual rate. Yield to maturity is the single discount rate that equates the present value of all cash flows to price. Current yield is annual coupon ÷ price. To compare rates, convert them to the same compounding frequency using (1 + r/m)^m, or e^r for continuous compounding.

Understand Yield Measures and Compounding

A yield is a rate of return stated per year. The same economic return can be quoted in different ways, and the quote depends on how often interest is compounded. A rate of 6% compounded semiannually is not the same number as 6% compounded annually, even though both use the word 6%.

Yield to maturity (YTM) is the single discount rate that makes the present value of a bond's promised cash flows equal to its market price. It assumes you hold to maturity, the issuer pays in full, and you can reinvest every coupon at the YTM. It is usually quoted as a rate with the bond's payment frequency, for example a semiannual-pay bond quotes a rate compounded twice a year.

Current yield is only annual coupon divided by price. It ignores the gain or loss from price moving to par at maturity and ignores time value. A discount bond has a YTM above its current yield, and a premium bond has a YTM below it. For a par bond, coupon rate, current yield and YTM are equal.

The bond-equivalent yield (BEY) is the semiannual yield multiplied by 2. It is a simple annualisation with no compounding within the year. The effective annual rate (EAR) includes compounding: EAR = (1 + BEY/2)² − 1. So EAR is always higher than BEY when the rate is positive.

To compare or convert rates, find the growth over one year. With m compounding periods per year, one year grows by (1 + r/m)^m. With continuous compounding it grows by e^r. Two rates are equivalent when they produce the same one-year growth factor. Continuous compounding is the limit as m goes to infinity and gives the highest growth for a given quoted rate.

Key formulas to remember

Bond price from YTM: P = Σ C/(1 + y/m)^t + F/(1 + y/m)^N; m = payments per year, t counts periods, N = total periods, C = coupon per period, F = face value. Solve for y with a financial calculator.
Current yield: Current yield = annual coupon ÷ market price; Uses the clean market price. Ignores capital gain or loss and time value.
Bond-equivalent yield: BEY = 2 × semiannual yield; Simple annualisation, no compounding within the year.
Effective annual rate: EAR = (1 + r/m)^m − 1; For a BEY, use m = 2 and r = BEY.
Continuous compounding: EAR = e^r − 1; r is the continuously compounded rate.
Discrete to continuous: r_c = m × ln(1 + r_m/m); Converts a rate with m periods per year to its continuous equivalent.
Continuous to discrete: r_m = m × (e^(r_c/m) − 1); Converts a continuous rate to a rate compounded m times a year.
Future value: FV = PV × (1 + r/m)^(mT) or PV × e^(rT); T is in years.

How to solve Yield Measures and Compounding questions

Use this routine for any yield or compounding question.

1Identify what is given: price, coupon, face value, maturity, payment frequency, or a quoted rate with its compounding frequency.
2Identify what is asked: YTM, current yield, BEY, EAR, or a rate under a different compounding basis.
3For yield measures, set up periods and period cash flows: period coupon = annual coupon ÷ m and N = years × m.
4For conversions, compute the one-year growth factor of the given rate: (1 + r/m)^m or e^r.
5Set that growth factor equal to the target form and solve for the target rate. For continuous, use ln; for discrete, use the m-th root.
6Annualise correctly: multiply a periodic rate by m for a quoted rate, or compound it for an effective rate.
7Check reasonableness: continuous rate < annual-compounded quoted rate, EAR > BEY, discount bond YTM > current yield > coupon rate.

Quickest way: Growth factor matching and ranking checks

When to use it: Use for MCQs that convert between compounding bases or ask you to compare yields.

Compute the one-year growth factor of the stated rate first. Everything else follows from it.
For discrete to continuous, use r_c = m × ln(1 + r/m). For small rates, expect r_c to be slightly below r.
Use ordering to eliminate options: continuous < semiannual < annual quote for the same effective return.
For YTM, use the calculator: N, PV = −price, PMT = period coupon, FV = face, then CPT I/Y and multiply by m.
For bond comparisons, check the sign: price below par means YTM > coupon rate and current yield sits between.

Common mistakes in Yield Measures and Compounding

Treating BEY as an effective annual rate.
Both are annual numbers, so they look interchangeable.
Fix: BEY = 2 × semiannual rate. To get EAR, compound: (1 + BEY/2)² − 1.
Entering the annual coupon and annual years into the calculator for a semiannual bond.
Students skip the step of adjusting for payment frequency.
Fix: Use N = years × 2 and PMT = annual coupon ÷ 2. Multiply the resulting I/Y by 2 to quote YTM.
Using current yield as the return on a bond.
It is simple and appears in quotes.
Fix: Remember it ignores the pull to par. Use YTM for total promised return.
Using ln(1 + r) when the rate is compounded m times a year.
Students memorise the annual case only.
Fix: Use r_c = m × ln(1 + r/m). Only for m = 1 does it reduce to ln(1 + r).
Forgetting to divide the rate by m before compounding, or forgetting the exponent m.
Rushing with formula (1 + r/m)^m.
Fix: Write both the period rate r/m and the number of periods m explicitly.
Comparing rates quoted on different bases directly.
Options present different compounding labels with similar numbers.
Fix: Convert all rates to EAR, or to one common basis, before ranking.

Worked examples

Example 1

A rate is quoted as 8% per year compounded semiannually. Find (a) the effective annual rate and (b) the equivalent continuously compounded rate.

Show the solution

Semiannual period rate = 8% ÷ 2 = 4% = 0.04.
(a) EAR = (1.04)² − 1 = 1.0816 − 1 = 0.0816, or 8.16%.
(b) r_c = 2 × ln(1.04). ln(1.04) = 0.039221.
r_c = 2 × 0.039221 = 0.078442, or about 7.84%.
Check: e^0.078442 = 1.0816, which matches the one-year growth factor.

Answer: EAR = 8.16% and continuously compounded rate ≈ 7.84%.

Example 2

A 5-year bond with a 6% annual coupon paid semiannually and face value 100 trades at 104.376. Find the current yield, and show that the YTM (bond-equivalent) is about 5%.

Show the solution

Annual coupon = 6. Current yield = 6 ÷ 104.376 = 0.05748, or 5.75%.
Semiannual coupon = 3, N = 10 periods, face = 100.
Test a semiannual yield of 2.5% (BEY 5%): annuity factor = (1 − 1.025^−10) ÷ 0.025. 1.025^10 = 1.28008, so 1.025^−10 = 0.78120.
Annuity factor = (1 − 0.78120) ÷ 0.025 = 8.7521. PV of coupons = 3 × 8.7521 = 26.256.
PV of face = 100 × 0.78120 = 78.120.
Price = 26.256 + 78.120 = 104.376, which matches the quoted price.
So the semiannual yield is 2.5% and YTM as a bond-equivalent yield = 2 × 2.5% = 5.00%.
Ordering check: coupon rate 6% > current yield 5.75% > YTM 5%, as expected for a premium bond.

Answer: Current yield ≈ 5.75%; YTM = 5.00% (BEY), below the current yield because the bond trades at a premium.

Exam tips

Always read how the rate is compounded before touching a calculator. The label is often the whole question.
Memorise the ordering for a premium bond: coupon rate > current yield > YTM. Reverse it for a discount bond.
On calculator questions, set payments per year to 1 and work in periods to avoid P/Y errors. Multiply I/Y by m afterwards.
For conversions, keep at least five decimals in ln and exponentials. Options are often close together.
If options include both the EAR and the BEY, check which one the question names before choosing.

Practice questions from Bond Yields and Return Calculations

Yield Measures and Compounding in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Yield Measures and Compounding: frequently asked questions

What is the difference between YTM and current yield?

YTM is the discount rate that equates all promised cash flows to the price, so it includes coupons, the price move to par and time value. Current yield is just annual coupon ÷ price. They are equal only when the bond trades at par.

How do I convert semiannual compounding to continuous compounding?

Use r_c = 2 × ln(1 + r/2), where r is the quoted semiannual rate. For example, 6% semiannual gives 2 × ln(1.03) = 5.91%. The continuous rate is always a little lower for the same effective return.

What is the difference between effective annual rate and bond-equivalent yield?

BEY doubles the semiannual yield and ignores compounding within the year. EAR compounds the semiannual yield: (1 + BEY/2)² − 1. EAR is higher whenever the yield is positive.

Does YTM assume anything about reinvestment?

Yes. YTM assumes you hold the bond to maturity, receive all payments, and reinvest coupons at the YTM itself. If reinvestment rates differ, your realised return will differ.